From [1]} it follows that there exists a constant \(C=C(D,\epsilon ,R)>0\)  such that for any linear wave \(\psi _{\ge 1}\)  that is supported on angular frequencies \(\ge 1\) , we have the following:
\(\int _{\Sigma _u} J^N[\psi _{\ge 1}]\cdot n_u\,d\mu _{\Sigma _u}\le C u^{-4+\epsilon } E_1[\psi _{\ge 1}],\) 
